Let X∼Gamma(α,β)X \sim \text{Gamma}(\alpha, \beta)X∼Gamma(α,β). What is the MGF MX(t)M_X(t)MX(t)?
(ββ−t)α(\frac{\beta}{\beta-t})^\alpha(β−tβ)α
eαte^{\alpha t}eαt
(1−βt)−α(1 - \beta t)^{-\alpha}(1−βt)−α
βαβ−t\frac{\beta^\alpha}{\beta-t}β−tβα